System for soft symbol decoding MIMO log-map detection

ABSTRACT

A soft symbol decoder for use in a multiple input multiple output (MIMO) and OFDM (orthogonal frequency division multiplexing) system. The decoder generates soft symbol values for a digital signal that represents a number of source bits. The source bits are transmitted as symbols in corresponding to points in a signaling constellation. Soft metrics are determined by searching for all possible multi-dimensional symbols that could have been transmitted. The method includes transmitting a sample of the multi-dimensional symbol using K transmit antennas. The multi-dimensional symbol is represent-able as a complex, K-dimensional vector x. Each vector component of vector x represents a signal transmitted with one of the K transmit antennas. After transmission through a communication channel, a sample corresponding to the transmitted sample is received. The received sample is represented by a complex, N-dimensional vector y, where N is the number of receive antennas in the MIMO system. After the sample is received, a soft metric L(b i ) is determined for each bit b i  encoded by x according to the equation: 
                 L   ⁡     (     b   i     )       =       σ     -   2       ·     (         min         x   j     ❘     b   i       =     -   1         ⁢            y   -     Hx   j            2       -       min         x   j     ❘     b   i       =     +   1         ⁢            y   -     Hx   j            2         )         ,   ,         
and x j  represents all possible values for x. In addition, a reduced complexity method is used for providing soft metric values in the MIMO system. This exemplary aspect reduces the complexity of the above computations from 2 BK  to 2 B(K−1) , where B is the number of bits transmitted per symbol per antenna.

CROSS-REFERENCE

This application is a continuation of U.S. patent application Ser. No.11/965,511, now U.S. Pat. No. 7,649,966, filed Dec. 27, 2007, entitled“SYSTEM FOR SOFT SYMBOL DECODING MIMO LOG-MAP DETECTION”, which is acontinuation application of U.S. patent application Ser. No. 10/376,060,now U.S. Pat. No. 7,315,576, filed Feb. 26, 2003, entitled “SYSTEM FORSOFT SYMBOL DECODING MIMO LOG-MAP DETECTION”, which is a continuation ofU.S. patent application Ser. No. 10/068,571, filed Feb. 5, 2002,entitled “SYSTEM FOR SOFT SYMBOL DECODING MIMO LOG-MAP DETECTION, nowabandoned. The entireties of these applications are hereby incorporatedby reference.

BACKGROUND OF THE INVENTION

The present invention relates generally to the field of communicationsystems and more specifically to systems for soft decoding data symbolsin multiple input multiple output communication systems.

Decoding is used by receivers to correctly identify transmitted datasymbols. During data transmission, noise from many sources may corrupttransmitted signals so that what is received is not necessarily what wastransmitted. For example, a signal transmitted as a 1 may be corruptedby noise and thereafter misinterpreted as a 0 at the receiver. Accurateinformation transmission depends upon being able to reliably detecttransmitted 1s as 1s and 0s as 0s.

One technique for transmitting data symbols is known as M-ary signaling.In M-ary signaling, the transmitter repeatedly selects one of M possibleunique symbols to send to represent the data being sent. In manysystems, M=2^(k) for some integer k and thus each symbol represents kbits. BPSK (binary phase shift keying) is an example of transmittingwith M=2; the symbols can be represented as 0 and 1. QPSK (quadraturephase shift keying) provides for four symbols where each symbol codesfor two bits (representable as 00, 01, 10 and 11).

In more complex keying schemes (e.g. QPSK), the signal amplitude variesas well as the phase. A signaling constellation is a mapping of symbolsto particular (amplitude, phase) pairs. For single transmit and receiveantennas, the signaling constellation has constellation pointscorresponding to the total number of symbols available. For instance, inQPSK, each symbol 00, 01, 10 and 11 is uniquely identifiable by aconstellation point in the signaling constellation. The problem ofsignal fading manifests itself as distortions of signalingconstellations where some of the points move closer together. Forexample, point 00 may be misinterpreted as point 01. This causes errorsat the receiver during the detection process, where a constellationpoint is misinterpreted as a different constellation point.

To facilitate correct interpretation of constellation points, softdecision decoders are employed at receivers for providing softdecisions. A “soft decision” is neither a 0 nor a 1 but a number inbetween that provides a degree of confidence as to whether the signal isa 0 or a 1. After soft decisions are made, the numbers are passed on toan apparatus or process that determines, from the context, whether thesignals are 0 or 1. One such process is the well-known Viterbialgorithm.

A system where one transmitter is used to transmit the data to onereceiver is referred to as a SISO (single input single output) system,whereas a system where more than one transmitter is used and more thanone receiver is used is referred to as a MIMO (multiple input multipleoutput) system. MIMO systems have advantages, such as being able toovercome the information capacity of SISO systems, increasing data rate,and obtaining increased receiver sensitivity. However, SISO systems havean advantage of simpler constellations. Conventional soft symboldecoding techniques used for SISO systems are inapplicable to MIMOsystems which have more complex constellations.

What is needed is a soft symbol decoding system that can reliablyperform soft symbol decoding for MIMO systems.

BRIEF SUMMARY OF THE INVENTION

A method for providing soft metrics in a MIMO communication system isdisclosed. The method provides soft metrics by searching for allpossible multi-dimensional symbols that could have been transmitted. Asnoted, such symbols are transmitted by K transmit antennas and receivedby N receive antennas, where K and N may be 2, 3, 4 . . . etc. Eachcomponent of the multi-dimensional symbol is transmitted with one of theK transmit antennas. The challenge is to ascertain that componentsreceived by the N receive antennas are what was transmitted by the Ktransmit antennas.

To accomplish this, soft metrics are provided to indicate likelihoodsthat a particular multi-dimensional symbol was transmitted.Specifically, a likelihood value is generated for each bit of themulti-dimensional symbol. Once generated, these metrics are forwarded toa decoder (e.g. a Viterbi decoder) for making hard decisions for thereceived symbols.

According to a first aspect of the present invention, the methodincludes the step of transmitting a sample of the multi-dimensionalsymbol using K transmit antennas. It should be noted that themulti-dimensional symbol is represent-able as a complex, K-dimensionalvector x. Each component of vector x represents a signal transmittedwith one of the K transmit antennas. After traveling through acommunication channel, the sample is received by the receiver. Thereceived sample is represented by a complex, N-dimensional vector y,where N is the number of receive antennas in the MIMO communicationsystem.

In one aspect, the vector y is related to vector x according to anequation: y=Hx+n, where H is an N×K channel matrix modeling phase shiftsand amplitude changes on paths between the K transmit and N receiveantennas, and n is a complex N-dimensional vector representingcommunication channel noise. After the sample is received, the methodfurther includes the step of determining a soft metric L(b_(i)) for eachbit b_(i) encoded by x according to the equation:

${{L\left( b_{i} \right)} = {\sigma^{- 2} \cdot \left( {{\min\limits_{{x_{j}❘b_{i}} = {- 1}}{{y - {Hx}_{j}}}^{2}} - {\min\limits_{{x_{j}❘b_{i}} = {+ 1}}{{y - {Hx}_{j}}}^{2}}} \right)}},$where x_(j) represents all possible values for x, and σ is a standarddeviation of the noise and/or interference in the received signal,assumed equal across antennas. σ can be estimated or some fixed valuecan be used if it is expected that all symbols have the samesignal-to-noise ratio. It should be observed that the complexity of thistechnique of soft value computation is 2^(BK) where B is the number ofbits transmitted per symbol on each antenna. For example, where K=2 andeach symbol is modulated using 16-QAM, for example, then themulti-dimensional symbol has 16²=256 possible values. When soft metricsare determined, they are passed to a Viterbi decoder that rendersrequisite hard decisions for the symbol bits.

According to another aspect of this invention, a reduced complexitymethod for providing soft metric values in the MIMO communication systemis taught. This method reduces the complexity of the above computationsfrom 2^(BK) to 2^(B(K−1)). For example, if K=2, and 64 QAM is employedfor the MIMO system, a 64-fold reduction in complexity is obtained. Inparticular, for small values of K e.g. 2 or 3, the reduction incomplexity is considerable.

Among other steps, the method includes providing a soft metric L(b_(i))for each bit b_(i) encoded by a complex vector x. For illustrativepurposes only, it is assumed that complex vector x contains no more thantwo elements namely a first element and a second element. This is thecase where two transmit antennas (K=2) are employed within the system.However, one of ordinary skill in the art will realize that the complexvector x may contain two or more vector components as proves necessaryin the present invention.

Next, this reduced complexity method includes the step of receiving asample represented by a complex vector y. Thereafter, a vector estimateof vector x is obtained by using the received vector y. In one aspect,this estimate is a scalar number obtained by factoring in amplitude andphase shifts in the communication channel. After the vector estimate ofvector x is determined, its elements i.e., the first and second elementsare obtained. For the first element, a hard decision is made on valuesfor each of its bits. Using these values, a soft metric L(b_(i)) foreach bit b_(i) encoded by the second element is determined. Similarly,for the second element, a hard decision is made on values for each ofits bits. Using these values, a soft metric L(b_(i)) is determined foreach bit b_(i) encoded by the first element. These soft metrics are thenforwarded to a Viterbi decoder where hard decisions are made.

A further understanding of the nature and advantages of the presentinvention herein may be realized by reference to the remaining portionsof the specification and the attached drawings. References to “steps” ofthe present invention should not be construed as limited to “step plusfunction” means, and is not intended to refer to a specific order forimplementing the invention. Further features and advantages of thepresent invention, as well as the structure and operation of variousembodiments of the present invention, are described in detail below withrespect to the accompanying drawings. In the drawings, the samereference numbers indicate identical or functionally similar elements.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a MIMO communication system in which soft symbol decoding isimplemented in accordance with a first embodiment of the presentinvention.

FIG. 2 is a method for providing soft symbol decoding ofmulti-dimensional symbols in accordance with one exemplary embodiment ofthe present invention.

DETAILED DESCRIPTION OF THE INVENTION

The following notation is used for number representation in theSpecification: (1) lower case italic represents scalar quantities; (2)lower case bold represents vectors, and (3) upper case bold is used formatrices.

Briefly, one aspect of the present invention provides for a soft symboldecoder in a multiple input multiple output (MIMO) communication system.The decoder receives signals and generates corresponding soft symbolvalues. Soft symbols are used for indicating the likelihood that eachbit of a symbol has a 0 or 1 state. That is, the probability that thetransmitted bit is 0 or 1.

Transmitted signals are represented as multi-dimensional symbols orvectors in a signaling constellation. Multi-dimensional symbols may besequential groupings of one-dimensional symbols. An example of suchone-dimensional symbols is a QAM constellation. Although each symbol hasin-phase and quadrature components, it is nevertheless considered acomplex one-dimensional symbol. Multidimensional vectors may containsequential groupings of such QAM constellation members. Each signalrepresenting a vector element is accurately detected by using maximum aposteriori probability as further described herein.

FIG. 1 is a block diagram of a communication system 100 in which softsymbol decoding is used in accordance with a first embodiment of thisinvention.

Among other components, communication system 100 includes a transmittersystem 122 and a receiver system 124. Transmitter system 124 comprises aforward error correction (FEC) encoder 104 for encoding an input datastream using a suitable FEC code, and an interleaver 105 for groupingthe input data for mapping. Transmitter system 124 also includesmodulator 106 for modulating the data stream into a form suitable fortransmission over a transmission channel 110 via multiple transmitantennas K.

Receiver system 124 includes a demodulator 114 for demodulating themodulated signal received via multiple receive antennas N, and a log-mapdecoder 120 for providing soft symbol decision values in accordance withan exemplary embodiment of the this invention. Receiver system 124further includes a de-interleaver 121 for inverting the interleavingintroduced at the transmitter, and a Viterbi decoder 116 for decodingthe soft decision symbol values into output data 130, as shown. Althoughnot shown, communication system 100 comprises other components such as ascrambler and FFT/IFFT processing blocks. Discussion of such componentsis omitted as not being critical to understanding this invention.

Communication system 100 is preferably an OFDM (orthogonal frequencydivision multiplexing) system. By using OFDM, signals are split intoseveral narrowband channels at different frequencies and processed tominimize interference between the channels. Preferably, OFDM is combinedwith a MIMO system having K transmit antennas and N receive antennas,where K and N can be greater than one. As noted, data rate and receiversensitivity are increased by using MIMO systems. In addition,communication system 100 is operable in various modes, one of which iscompliant with the IEEE 802.11a standard entitled “Part 11: Wireless LANMedium Access Control (MAC) and Physical Layer (PHY) specifications,High-speed Physical Layer in the 5 GHz Band.”

As shown in FIG. 1, input data 102 is fed into FEC encoder 104, wherethe data stream is encoded. The purpose of encoding is to decrease thebit error rate of the transmitted signal by introducing some form ofredundancy into the transmitted data stream. After input data 102 isencoded, the encoded data bits are fed into an interleaver 105. Thesebits are grouped by interleaver 105 into groups that can be directlymapped to symbols. After interleaving, the grouped bits are provided toconstellation mapper (not shown) which maps the encoded bits intosignaling constellation symbols. Bits can be mapped onto BPSK, QPSK, 16QAM and 64 QAM tones, for example. Thereafter, the complex numbersymbols are modulated by modulator 106 for transmission via the Ktransmit antennas. Note that data bits are transmitted as K dimensionalvectors containing K complex number elements. Each vector element ischosen from the M-ary constellation used. For example, if K=2, and 16QAM is used, vector elements may be represented by 0001 and 1010.Further, one element is transmitted by the first antenna while the otheris transmitted by the second antenna. Also, note that total bitstransmitted per channel usage is given by K*k=8.

At receiver 124, the transmitted signal is received and the reverse ofthe transmitter process occurs. However, in accordance with thisinvention, soft symbol decoding is performed by log map decoder 120.This soft metric is then fed to Viterbi decoder 116. The soft metricsare for indicating the likelihood of each state of transmitted bits.That is, the probability that the transmitted bit is 0 or 1. Thisprobability is derived by obtaining the squared error between thereceived signal and all possible candidate transmissions. Log mapdecoder 120 calculates the difference of minimum distances between thereceived symbol and all possible symbols that have been sent with a 0 or1 for each bit location. A normalized version of this distance is thedesired soft metric. One of ordinary skill in the art will realize thatlog map decoder 120 comprises logic circuitry implementable throughsoftware, hardware or a combination of both. For example, one or moresoftware instructions may be provided for running the steps of thisinvention.

Note that in the MIMO system used in the present invention, there are atleast 256 possible candidate transmissions in the present invention.That is, where K=2, and each element of the transmitted vector ismodulated using 16 QAM for example, there are 16²=256 possibletransmissions. This means that decoding computations are 16 times ascomplex relative to SISO systems. In general, there will be M^(K)possible candidate transmissions, where M is the signaling constellationused. Also, the complexity of this way of soft value computation is2^(BK), where B is the number of bits transmitted per symbol on eachantenna.

FIG. 2 is a method 200 for providing soft symbol decoding in MIMOcommunication system 100 in accordance with one exemplary embodiment ofthe present invention.

Specifically, soft values for a received multi-dimensional symbol isprovided by Log MAP decoder 120. These values are thereafter forwardedto FEC decoder 120 a. As noted, K transmit antennas and N receiveantennas are employed in MIMO communication system 100. One objective isto combine the increased data rate due to the use of multiple transmitantenna, and the increased receiver sensitivity (i.e. lower requiredsignal to noise ratio) of obtained by using multiple receivers.

At block 302, a sample of a multi-dimensional symbol is transmitted bytransmit antenna K. The multi-dimensional symbol is represent-able as aK-dimensional vector x containing K complex numbers x₁ . . . x_(K). Eachcomplex number is chosen from an M-ary constellation, where M is thenumber of bits encoded. For example, in a 16-QAM constellation, each ofthe complex numbers is one of sixteen constellation points. Note thatk=4 bits so that each complex number encodes 4 bits. Thus, each vectorcomponent of the vector x represents a signal transmitted with one ofthe K transmit antenna. In total K×k bits are transmitted.

At block 304, the signal is transmitted through transmission channel110. This channel is represented by the equation:y=Hx+n  (1)

where y is the symbol received by receiver 124, y being an N-dimensionalcomplex vector, H is an N×K vector channel matrix consisting of Kcolumns vectors h₁ . . . h_(K). Each of these vectors h_(k) representsthe channel coefficients modeling phase shift and amplitude change onthe path between the K-th transmit antenna to the N receive antennas 1 .. . N. In addition, n is a complex, N-dimensional vector representingthe channel noise picked up at the N-th receive antenna. Each element nhas the same expected power E{n_(i)*n_(i)}=E[|n_(i)|²]=σ².

At block 306, a sample represented by the complex vector y is receivedby receiver 124. Thereafter, log map decoder 120 applies soft decisiondecoding for each bit b_(i) encoded by vector x according to the sameequation:

$\begin{matrix}{{L\left( b_{i} \right)} = {\sigma^{- 2} \cdot \left( {{\min\limits_{{x_{j}❘b_{i}} = {- 1}}{{y - {Hx}_{j}}}^{2}} - {\min\limits_{{x_{j}❘b_{i}} = {+ 1}}{{y - {Hx}_{j}}}^{2}}} \right)}} & (2)\end{matrix}$

where x_(j) are all possible values for x. The receiver has to searchover all possible x which are vectors. So if K=2 and each element of xis modulated using 16-QAM, for example, then x has 16²=256 possiblevalues. In general the complexity of this way of soft value computationis 2^(BK), where B is the number of bits transmitted per symbol on eachantenna, and K is the number of antennas.

It should be noted that for any N-dimensional complex vector v, thesquared vector length is calculated as

$\begin{matrix}{{v} = {\sqrt{v^{*} \cdot v} = \sqrt{\sum\limits_{i = 1}^{N}\;{v_{i}^{*} \cdot v_{i}}}}} & (3)\end{matrix}$Thus, in terms of real and imaginary components the complex elements ofv:v _(i) *·v _(i) =|v _(i)|² =Re(v _(i))² +Im(v _(i))²  (4)This vector length can be approximated (with minimal soft value accuracydegradation) as

$\begin{matrix}{{v} \approx {{\sum\limits_{i = 1}^{N}\;{{{Re}\left( v_{i} \right)}}} + {{{Im}\left( v_{i} \right)}}}} & (5)\end{matrix}$

Reduced Complexity MIMO Reception

This exemplary embodiment is a method that reduces the complexity of theabove computations from 2^(BK) to 2^(B(K−1)). For small values of K,i.e. 2 or 3 transmit antennas the reduction in complexity isconsiderable, especially if B is large. For instance for 64 QAM and K=2,a 64-fold reduction is obtained.

When y is received symbol, and H=[h₁ . . . h_(K)] is the channel matrix,first a scalar number z is computed as follows.

$\begin{matrix}{z = {\frac{h_{1}^{*}y}{h_{1}^{*}h_{1}} = \frac{h_{1}^{*}y}{{h_{1}}^{2}}}} & (6)\end{matrix}$Note that z is obtained by dividing two scalars. Here h₁* is the complexconjugate transpose of the first column of channel matrix H. Thereafter,the receiver starts a loop for all values of x₂, . . . X_(K). Thereforethis loop is executed 2^(B(K−1)) times.

Loop:

An estimate of x₁, the first element of the transmitted vector x isformed as follows, by correcting x

$\begin{matrix}{{\hat{x}}_{1} = {z - \frac{{h_{1}^{*}h_{2}x_{2}} - \ldots - {h_{1}^{*}h_{k}x_{k}}}{h_{1}^{*}h_{1}}}} & (7)\end{matrix}$

A simple QAM slicer is now applied to {circumflex over (x)}₁ to makehard decisions on the values of the bits of symbol x₁. Then thehard-decided x₁, and the x₂ . . . x_(N) loop variables are substitutedin the equation below:∥y−Hx_(j)∥²  (8)

End Loop

The 2^(B(K−1)) values obtained are used to determine the minima in theequation below

$\begin{matrix}{{L\left( b_{i} \right)} = {\sigma^{- 2} \cdot \left( {{\min\limits_{{x_{j}❘b_{i}} = {- 1}}{{y - {Hx}_{j}}}^{2}} - {\min\limits_{{x_{j}❘b_{i}} = {+ 1}}{{y - {Hx}_{j}}}^{2}}} \right)}} & (9)\end{matrix}$and obtain the L(b_(i)) values for all bits in symbols x₂ . . . x_(K).The entire procedure above is now redone, except that:

$\begin{matrix}{z = {\frac{h_{K}^{*}y}{h_{K}^{*}h_{K}} = \frac{h_{K}^{*}y}{{h_{K}}^{2}}}} & (10)\end{matrix}$

This yields soft values for all bits in symbols x₁ . . . x_(K−1). Thesoft values for x₂ . . . x_(K−1) can now be thrown away since they havealready been obtained in the first step. Those for x₁ will be retainedand added to the values for x₂ . . . x_(K) obtained before.

While the above is a complete description of exemplary specificembodiments of the invention, additional embodiments are also possible.For example, a simplified variation of the reduced complexity algorithmis possible that only does the search loop once, possibly withperformance costs. In contrast, the above technique does the loop twiceto come up with the soft values for the first antenna. Rather thanmaking only hard decisions on the values of the bits of symbol x₁, aconventional QAM soft decision is made on

${\hat{x}}_{1} = {z - \frac{{h_{1}^{*}h_{2}x_{2}} - \ldots - {h_{1}^{*}h_{K}x_{K}}}{h_{1}^{*}h_{1}}}$to obtain a soft metric values. Another augmentation to determiningL(b_(i)) would include determining soft metrics with the addedassumption that the noise variance is not spatially white and iscorrelated. That is, the noise variance varies across receive antennas.The soft-metric computation would then be modified to be

$\begin{matrix}{{L\left( b_{i} \right)} = \left( {{\min\limits_{{x_{j}❘b_{i}} = {- 1}}{{\left( {y - {Hx}_{j}} \right){R^{- 1}\left( {y - {Hx}_{j}} \right)}^{*}}}^{2}} - {\min\limits_{{x_{j}❘b_{i}} = {+ 1}}{{\left( {y - {Hx}_{j}} \right){R^{- 1}\left( {y - {Hx}_{j}} \right)}^{*}}}^{2}}} \right)} & (11)\end{matrix}$

where R−1 is the inverse of the spatial noise/interferenceauto-covariance matrix. In the case that the noise is independent acrossantennas (but not necessarily equal), this inverse matrix can bedescribed for the N=3 case as:

$R^{- 1} = \begin{bmatrix}{1/\sigma_{1}^{2}} & 0 & 0 \\0 & {1/\sigma_{2}^{2}} & 0 \\0 & 0 & {1/\sigma_{3}^{2}}\end{bmatrix}$

where σ_(i) ² is the noise/interference power at antenna input i. Thus,the above description should not be taken as limiting the scope of theinvention, which is defined by the appended claims along with their fullscope of equivalents.

Appendix

Pseudo-code for both exhaustive search and reduced complexity algorithmsare included below.

// variable declaration and initialization #define B 6 // number of bitsper antenna per symbol #define M 2 // number of transmit antennas doublemin [2] [M] [B]; // stores minima double L [M] [B]; // log likelihoodratios for (m=1 .. M) for (b=1 .. B) for (v=0, 1)  min [v] [v] [m] = ∞;// pseudo code for log-MAP decoder for (bits[1] = 0 .. 2^(B)-1,bits[2]=0 .. 2^(B)-1, ..., bits[M]=0 .. 2^(B)-1) { dist = || r - H[1]*map(bits[1]) - ... - H[M] *map(bits[M]) ||²; // note: map ( ) maps bitsto constellations for (m = 1 .. M), for (b = 0 .. B-1) if (min [bits [m]& (1<<b)] [m] [b] > dist) min [bits [m] & (1<<b)] [m] [b] = dist; }for(m= 1 .. M) for (b = 0 .. B-1) L [m] [b] = min [0] [m] [b] − min [1][m] (b]; // pseudo code for Reduced complexity log-MAP decoder z = H[1]′*r; // max ratio combining of x1 for (bits[2] = 0 .. 2^(B)-1) { // note:bits[1] excluded from loop! bits[1] = slice (z - H[1]′ *H[2]*map(bits[2])) dist = || r - H[1] *map(bits[1]) ||²; this is anadditional simplification, it is not essential for the technique for(m=1 .. M) for (b = 0 .. B-1) if(min [bits [m] & (1 <<b)] [m] [b] > dist) min [bits [m] & (1 <<b)] [m] [b] = dist; } z = H[2]′ *r; //max ratiocombining of x2 for(bits[1]=0 .. 2^(B)-1) { // note: bits[1] excludedfrom loop! bits[2] = slice(z - H[2]′ *H[1] *map(bits[1])) dist = || r -H[2] *map(bits[2]) ||²; for(m= 1 .. M) for(b = 0 .. B-1) if (min [bits[m] & (1<<b)] [m] [b] > dist) min [bits [m] & (1<<b)] [m] [b] = dist; }for(m= 1 .. M) for(b = 0 .. B-1) L[m] [b] = min[0] [m] [b] − min[1] [m][b];

What is claimed is:
 1. A communications apparatus, comprising: an inputconfigured to receive a digital signal that represents a symboltransmitted over a transmit antenna; a first decoder configured toperform a computation of a set of soft metrics of the transmittedsymbol, wherein the computation has a complexity of 2^(B(K−1)) where Bis a number of bits transmitted per symbol on an antenna, and K is anumber of transmit antennas, the set of soft metrics to indicate alikelihood of a state of bits of the symbol, at least in part bycomputing an estimate of a first element of the transmitted symbol,based on the estimate, making a hard decision on values of bits of thefirst element to form a hard-decided first element, computing softmetrics for second to K^(th) elements of the transmitted symbol at leastpartly in terms of the hard-decided first element, computing a softmetric for at least the first element of the transmitted symbol, andobtaining the soft metrics of the transmitted symbol in terms of thesoft metric for the first element of the transmitted symbol and the softmetrics for the second to K^(th) elements; and a second decoderconfigured to utilize the set of soft metrics to determine a symbolvalue corresponding to the transmitted symbol.
 2. The communicationsapparatus of claim 1, wherein the symbol is multi-dimensional andrepresented as a dimensional vector comprising complex numbers.
 3. Thecommunications apparatus of claim 2, wherein a size of the dimensionalvector is based on a number of transmit antennas employed to transmitthe digital signal.
 4. The communications apparatus of claim 2, whereinthe complex numbers are selected from an M-ary constellation, where M isa number of bits encoded.
 5. A method for facilitating wirelesscommunication, comprising: receiving a symbol represented as a complexvector; applying a soft decision decoding for the symbol, wherein thesoft decision decoding includes computing an estimate of a first elementof the symbol, based on the estimate, making a hard decision on valuesof bits of the first element to form a hard-decided first element,computing soft metrics for second to K^(th) elements of the symbol atleast partly in terms of the hard-decided first element, where K is anumber of transmit antennas corresponding to the symbol, computing asoft metric for at least the first element of the symbol, and obtainingsoft metrics of the symbol in terms of the soft metric for the firstelement of the symbol and the soft metrics for the second to K^(th)elements; and utilizing the soft metrics of the symbol to determine asymbol value.
 6. The method of claim 5, further comprising ascertaininga likelihood of a state associated with at least one bit encoded by thecomplex vector, wherein the state is 0 or
 1. 7. The method of claim 6,wherein the ascertaining further comprising obtaining a squared errorbetween a signal that includes the symbol and possible candidatesignals.
 8. A computer program product for a wireless communicationnetwork, comprising: a computer-readable non-transitory mediumcomprising: code for receiving at least one symbol represented as acomplex vector; code for applying soft decision decoding for the complexvector, the soft decision decoding including computing an estimate of afirst element of the at least one symbol, based on the estimate, makinga hard decision on values of bits of the first element to form ahard-decided first element, computing soft metrics for second to K^(th)elements of the at least one symbol at least partly in terms of thehard-decided first element, where K is a number of transmit antennascorresponding to the at least one symbol, computing a soft metric for atleast the first element of the at least one symbol, and obtaining softmetrics of the at least one symbol in terms of the soft metric for thefirst element of the at least one symbol and the soft metrics for thesecond to K^(th) elements; and code for employing the soft metrics todetermine at least one symbol value of the at least one symbol.
 9. Thecomputer program product of claim 8, the computer-readablenon-transitory medium further comprising code for determining likelihoodstates associated with bits encoded by the complex vector, wherein thelikelihood states are 0 or
 1. 10. The computer program product of claim9, the computer-readable non-transitory medium further comprising codefor obtaining a squared error between signals including the at least onesymbol.
 11. Communications means, comprising: input means for receivinga digital signal that represents a symbol transmitted over a transmitantenna; first decoder means for performing a computation of a set ofsoft metrics of the transmitted symbol, wherein the computation has acomplexity of 2^(B(K−1)), where B is a number of bits transmitted persymbol on an antenna, and K is a number of transmit antennas, the set ofsoft metrics to indicate a likelihood of a state of bits of the symbol,the first decoder means including estimating means for computing anestimate of a first element of the transmitted symbol, decision meansfor making a hard decision on values of bits of the first element toform a hard-decided first element, and metrics computing means forcomputing soft metrics for second to K^(th) elements of the transmittedsymbol at least partly in terms of the hard-decided first element,computing a soft metric for at least the first element of thetransmitted symbol, and obtaining the soft metrics of the transmittedsymbol in terms of the soft metric for the first element of thetransmitted symbol and the soft metrics for the second to K^(th)elements; and second decoder means for utilizing the set of soft metricsto determine a symbol value corresponding to the transmitted symbol.